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Online Cryptography Course Dan Boneh Stream ciphers PRG Security Defs Dan Boneh n Let G:K ⟶ {0,1} be a PRG Goal: define what it means that is “indisHnguishable” from Dan Boneh StaHsHcal Tests Sta$s$cal test on {0,1}n: an alg A s.t A(x) outputs “0” or “1” Examples: Dan Boneh StaHsHcal Tests More examples: Dan Boneh Advantage Let G:K ⟶{0,1}n be a PRG and A a stat test on {0,1}n Define: A silly example: A(x) = 0 ⇒ AdvPRG [A,G] = 0 Dan Boneh Suppose G:K ⟶{0,1}n saHsfies msb(G(k)) = 1 for 2/3 of keys in K Define stat test A(x) as: if [ msb(x)=1 ] output “1” else output “0” Then AdvPRG [A,G] = | Pr[ A(G(k))=1] -‐ Pr[ A(r)=1 ] | = | 2/3 – 1/2 | = 1/6 Dan Boneh Secure PRGs: crypto definiHon Def: We say that G:K ⟶{0,1}n is a secure PRG if Are there provably secure PRGs? but we have heurisHc candidates Dan Boneh Easy fact: a secure PRG is unpredictable We show: PRG predictable ⇒ PRG is insecure Suppose A is an efficient algorithm s.t for non-‐negligible ε (e.g ε = 1/1000) Dan Boneh Easy fact: a secure PRG is unpredictable Define staHsHcal test B as: Dan Boneh Thm (Yao’82): an unpredictable PRG is secure Let G:K ⟶{0,1}n be PRG “Thm”: if ∀ i ∈ {0, … , n-‐1} PRG G is unpredictable at pos i then G is a secure PRG If next-‐bit predictors cannot disHnguish G from random then no staHsHcal test can !! Dan Boneh Let G:K ⟶{0,1}n be a PRG such that from the last n/2 bits of G(k) it is easy to compute the first n/2 bits Is G predictable for some i ∈ {0, … , n-‐1} ? Yes No More Generally Let P1 and P2 be two distribuHons over {0,1}n Def: We say that P1 and P2 are computa$onally indis$nguishable (denoted ) R Example: a PRG is secure if { k ⟵K : G(k) } ≈p uniform({0,1}n) Dan Boneh End of Segment Dan Boneh